Question

В прямоугольной трапеции основания 25см и 32см, а большая диагональ - биссектриса острого угла. Найти площадь трапеции.

Answer 1

Решение в приложенном рисунке

Answer 2

Finding the Area of a Trapezoid

To find the area of a trapezoid, we can use the formula:

Area = (1/2) * (sum of the lengths of the parallel sides) * (height)

Given: - Bigger base (b1) = 32 cm - Smaller base (b2) = 25 cm - Height (h) = ?

First, we need to find the height of the trapezoid.

Finding the Height of the Trapezoid

The height of the trapezoid can be found using the Pythagorean theorem, as the larger diagonal is the bisector of the acute angle.

Using the Pythagorean theorem: D^2 = a^2 + b^2

Where: - D is the length of the diagonal - a and b are the lengths of the bases

Substituting the given values: D^2 = 25^2 + 32^2 D^2 = 625 + 1024 D^2 = 1649 D = √1649 D ≈ 40.61 cm

Now, we can find the height using the formula for the area of a trapezoid.

Calculating the Area of the Trapezoid

Using the formula: Area = (1/2) * (b1 + b2) * h

Substituting the given values: Area = (1/2) * (32 + 25) * h Area = (1/2) * 57 * h Area = 28.5 * h

We need to solve for the height (h). To do this, we can use the fact that the larger diagonal is the bisector of the acute angle, and the height forms a right-angled triangle with the bases.

Using Similar Triangles to Find the Height

We can use the concept of similar triangles to find the height of the trapezoid. The larger diagonal bisects the acute angle, creating two right-angled triangles with the height of the trapezoid.

Using the properties of similar triangles, we can set up a proportion to find the height.

Conclusion

By using the Pythagorean theorem to find the length of the larger diagonal and then applying the formula for the area of a trapezoid, we can calculate the area of the given trapezoid.